Mayer–Vietoris Sequence for dihedral homology of commutative algebras

Document Type : Research

Authors

1 math. depatment, fucalty of science, Aswan university

2 Department of Mathematics, Faculty of Science, Aswan University

Abstract

Dihydral homology offers a powerful framework for analyzing commutative algebras endowed with symmetry, serving as a key link between homological algebra, algebraic geometry, and mathematical physics. By incorporating group actions—particularly dihedral symmetries—it extends classical Hochschild and cyclic homology, enabling the study of structured deformations, invariant theory, and equivariant algebraic structures. This theory not only enriches our understanding of symmetric algebras but also provides computational tools for geometric and physical applications, such as orbifold theories and deformation quantization. Further research into its properties could deepen insights into derived geometry, operadic methods, and topological field theories, revealing new connections across disciplines. In this work, we explore the foundational aspects of dihedral homology for commutative algebras, focusing on its defining features. We will introduce the Mayer–Vietoris sequence for dihedral homologies of commutative algebras with prove. Key topics include long exact sequences and Morita invariance, which mirror classical homological tools while adapting to symmetric constraints. We also discuss its geometric interpretations, such as ties to equivariant cohomology and moduli spaces with group actions. By unifying algebraic and geometric perspectives, dihydral homology not only advances pure mathematics but also opens doors to novel applications in mathematical physics, particularly in systems with discrete symmetries. This investigation aims to provide a concise yet comprehensive overview of its essential characteristics, paving the way for future theoretical and applied developments.

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Aswan Science and Technology Bulletin
Volume 3, Issue 1
June 2025
Pages 159-164

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